I'm interested in computer animation, and I'd like to work at a gaming company after I graduate. I've tried learning how to use computer modeling programs like 3dsmax and Maya through tutorials, and so far I am enjoying it. But i am concerned that companies won't want to hire me if I don't have any programming experience. Is it necessary to know programming in order to be successful in computer animation?
Nope, not at all, as far as I know. Games companies are usually very modular in their team setups (again, as far as I know). They have an art department, a design department, a programming department, etc...
The best thing you could do would be to phone up the HR department of the company you want to work at, and ask them what you would need to do to get them to hire you as an artist. My best guess would be a large and detailed portfolio of high-quality models. I don't think you'd need programming.
It's not necessary, but remember that the less you know about the other aspects of game development, the less "mobility" you have. In other words, the more well-rounded you are, the more marketable you are and can qualify for more positions.
With that said, however, it's fairly common to enter a game development company at the entry level (for example: 2D Artist Intern) and then learn the ropes of the other departments while you work. 2D art/texturing often leads to 3D modeling, leading to animation, leading to environment interaction/design, etc. You become an attractive candidate if you show the will to learn and expand.
The others touched on this pretty well. Learning a language like Lua, or even Maxscript if you're using 3ds Max, could help separate you from other people.
A highly polished portfolio is extremely important, ones with animations, short films, both high and low poly models, etc. Some studios will look to see how far 'out of the box' you can go with your animations and models.